Backward Ricci flow on locally homogeneous 3-manifolds
Backward Ricci flow on locally homogeneous 3-manifolds
复制标题
局部均匀 3 流形上的向后 Ricci 流
DOI：
10.4310/cag.2009.v17.n2.a6
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发表时间：
2009
影响因子：
0.7
通讯作者：
L. Saloff‐Coste
中科院分区：
文献类型：
--
作者：
Xiaodong Cao;L. Saloff‐Coste
在[6]中，Isenberg 和 Jackson 研究了齐次 3 流形上的 Ricci 流。由于齐次 3 流形是 3 流形几何化的模型和构建块，因此研究这种基本情况下各种几何流的行为是自然且重要的。参见 [2，第 1 章]。进一步的研究参见[5,7,8]。出于明显的原因，工作集中在 Ricci 流的前向行为上，尽管在均匀情况下，流简化为常微分方程组，并且不妨碍向后流的研究。在[3, 4]中，作者研究了均匀3流形上交叉曲率流的前向和后向极限。事实上，在交叉曲率流的情况下，并不完全清楚哪个方向更自然。 [4]中获得的结果表明，还应该研究 Ricci 流的后向行为，这就是本文的主题。回想一下，流形上的里奇流是满足方程的黎曼度量 g(t) 的流
In [6], Isenberg and Jackson studied the Ricci flow on homogeneous 3-manifolds. As homogeneous 3-manifolds are the models and building blocks of the geometrization of 3-manifolds, it is natural and important to study the behavior of various geometric flows in this basic case. See [2, Chapter 1]. Further studies are in [5, 7, 8]. For obvious reasons, works have focused on the forward behavior of the Ricci flow although, in the homogeneous case, the flow reduces to a system of ordinary differential equations and there is no obstruction to the study of the backward flow. In [3, 4], the authors studied the forward and backward limits of the cross curvature flow on homogeneous 3-manifolds. Indeed, in the case of the cross curvature flow it is not entirely clear which direction is more natural. The results obtained in [4] suggest that the backward behavior of the Ricci flow should be studied as well and this is the subject of this paper. Recall that the Ricci flow on a manifold is a flow of Riemannian metric g(t) satisfying the equation